Initially studied by Markov around \(1880\), the Lagrange and Markov spectra are complicated subsets of the real line that play a crucial role in the study of Diophantine approximation and the study of binary quadratic forms. In the \(1920\)s, Perron gave an amazingly useful description of the spectra in terms of continued fractions and, in the \(1960\)s, Freiman demonstrated that the Lagrange spectrum is a strict subset of the Markov spectrum. It still remains a difficult task to find points in the complement of the Lagrange spectrum within the Markov spectrum and modern research is focussed on further developing our understanding of this complement.

In this talk, I will introduce these spectra, discussing the historical results above, and speak about recent works with Harold Erazo, Carlos Matheus and Carlos Gustavo Moreira finding new points in the complement and obtaining better lower bounds for its Hausdorff dimension.

PDF abstract here.

A (2-dimensional) realisation of a graph G is a pair \((G,p)\), where \(p\) maps the vertices of \(G\) to \(\mathbb{R}^2\). A realisation is flexible if it can be continuously deformed while keeping the edge lengths fixed, and rigid otherwise. Similarly, a graph is flexible if its generic realisations are flexible, and rigid otherwise. We show that a minimally rigid graph has a flexible realisation with positive edge lengths if and only if it is not a 2-tree. This confirms a conjecture of Grasegger, Legersky and Schicho. Our proof is based on a characterisation of graphs with \(n\) vertices and \(2n-3\) edges and without stable cuts due to Le and Pfender. We also strengthen a result of Chen and Yu, who proved that every graph with at most \(2n-4\) edges has a stable cut, by showing that every flexible graph has a stable cut. Additionally, we investigate the number of NAC-colourings in various graphs. A NAC-colouring is a type of edge colouring introduced by Grasegger, Legersky and Schicho, who showed that the existence of such a colouring characterises the existence of a flexible realisation with positive edge lengths.

This is joint work with Clinch, Garamvolgyi, Haslegrave, Huynh and Legersky.

Folding of the ideal polygon in its \(j\)–th vertex reflects the vertex in the corresponding short diagonal. We show that compositions of such foldings along any Coxeter element provide Liouville integrable system on the moduli space of ideal polygons. This result also provides integrability for Shramm circle patterns. This is joint work with Anton Izosimov.

Let \(\delta(p)\) tend to zero arbitrarily slowly as \(p\to\infty\). We exhibit an explicit set \(\mathcal{S}\) of primes \(p\), defined in terms of simple functions of the prime factors of \(p-1\), for which the least primitive root of \(p\) is at most \( p^{1/4-\delta(p)}\) for all \(p\in \mathcal{S}\), where \(\#\{p\leq x: p\in \mathcal{S}\} \sim \pi(x)\) as \(x\to\infty\). This is a joint work with Kevin Ford and Andrew Granville.